Identities and generating functions on Chebyshev polynomials

In this article we present the classical theory of Chebyshev polynomials starting from the definition of a family of complex polynomials, including both the first and second kind classical Chebyshev polynomials, which are related to their real and imaginary parts. This point of view permits one to derive a lot of generating functions and relations between Chebyshev families of two kinds, which are essentially new, as exponential generating functions, bilinear and bilinear exponential generating functions. We also deduce relevant relations for products of Chebyshev polynomials and the related generating functions.